Uniqueness of some Calabi–Yau metrics on $${\mathbf {C}}^{{n}}$$

Abstract

We consider the Calabi–Yau metrics on $$\mathbf {C}^n$$ constructed recently by Yang Li, Conlon–Rochon, and the author, that have tangent cone $$\mathbf {C}\times A_1$$ at infinity for the $$(n-1)$$ -dimensional Stenzel cone $$A_1$$ . We show that up to scaling and isometry this Calabi–Yau metric on $$\mathbf {C}^n$$ is unique. We also discuss possible generalizations to other manifolds and tangent cones.